21 found
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  1.  39
    On Lovely Pairs of Geometric Structures.Alexander Berenstein & Evgueni Vassiliev - 2010 - Annals of Pure and Applied Logic 161 (7):866-878.
    We study the theory of lovely pairs of geometric structures, in particular o-minimal structures. We use the pairs to isolate a class of geometric structures called weakly locally modular which generalizes the class of linear structures in the settings of SU-rank one theories and o-minimal theories. For o-minimal theories, we use the Peterzil–Starchenko trichotomy theorem to characterize for a sufficiently general point, the local geometry around it in terms of the thorn U-rank of its type inside a lovely pair.
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  2.  24
    The Independence Property in Generalized Dense Pairs of Structures.Alexander Berenstein, Alf Dolich & Alf Onshuus - 2011 - Journal of Symbolic Logic 76 (2):391 - 404.
    We provide a general theorem implying that for a (strongly) dependent theory T the theory of sufficiently well-behaved pairs of models of T is again (strongly) dependent. We apply the theorem to the case of lovely pairs of thorn-rank one theories as well as to a setting of dense pairs of first-order topological theories.
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  3.  10
    Thorn Independence in the Field of Real Numbers with a Small Multiplicative Group.Alexander Berenstein, Clifton Ealy & Ayhan Günaydın - 2007 - Annals of Pure and Applied Logic 150 (1-3):1-18.
    We characterize þ-independence in a variety of structures, focusing on the field of real numbers expanded by predicate defining a dense multiplicative subgroup, G, satisfying the Mann property and whose pth powers are of finite index in G. We also show such structures are super-rosy and eliminate imaginaries up to codes for small sets.
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  4.  49
    Weakly One-Based Geometric Theories.Alexander Berenstein & Evgueni Vassiliev - 2012 - Journal of Symbolic Logic 77 (2):392-422.
    We study the class of weakly locally modular geometric theories introduced in [4], a common generalization of the classes of linear SU-rank 1 and linear o-minimal theories. We find new conditions equivalent to weak local modularity: "weak one-basedness", absence of type definable "almost quasidesigns", and "generic linearity". Among other things, we show that weak one-basedness is closed under reducts. We also show that the lovely pair expansion of a non-trivial weakly one-based ω-categorical geometric theory interprets an infinite vector space over (...)
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  5.  12
    Dense Codense Predicates and the NTP2.Alexander Berenstein & Hyeung-Joon Kim - 2016 - Mathematical Logic Quarterly 62 (1-2):16-24.
  6.  14
    Simple Stable Homogeneous Expansions of Hilbert Spaces.Alexander Berenstein & Steven Buechler - 2004 - Annals of Pure and Applied Logic 128 (1-3):75-101.
    We study simplicity and stability in some large strongly homogeneous expansions of Hilbert spaces. Our approach to simplicity is that of Buechler and Lessmann 69). All structures we consider are shown to have built-in canonical bases.
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  7.  15
    Hilbert Spaces Expanded with a Unitary Operator.Camilo Argoty & Alexander Berenstein - 2009 - Mathematical Logic Quarterly 55 (1):37-50.
    We study Hilbert spaces expanded with a unitary operator with a countable spectrum. We show that the theory of such a structure is ω -stable and admits quantifier elimination.
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  8.  28
    Hilbert Spaces with Generic Groups of Automorphisms.Alexander Berenstein - 2007 - Archive for Mathematical Logic 46 (3-4):289-299.
    Let G be a countable group. We prove that there is a model companion for the theory of Hilbert spaces with a group G of automorphisms. We use a theorem of Hulanicki to show that G is amenable if and only if the structure induced by countable copies of the regular representation of G is existentially closed.
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  9.  17
    Definable Subgroups of Measure Algebras.Alexander Berenstein - 2006 - Mathematical Logic Quarterly 52 (4):367-374.
    We show that type-definable subgroups of measure algebras are definable.
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  10.  27
    Imaginaries in Hilbert Spaces.Itay Ben-Yaacov & Alexander Berenstein - 2004 - Archive for Mathematical Logic 43 (4):459-466.
    We characterise imaginaries (up to interdefinability) in Hilbert spaces using a Galois theory for compact unitary groups.
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  11.  13
    Generic Trivializations of Geometric Theories.Alexander Berenstein & Evgueni Vassiliev - 2014 - Mathematical Logic Quarterly 60 (4-5):289-303.
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  12.  19
    Simple Stable Homogeneous Groups.Alexander Berenstein - 2003 - Journal of Symbolic Logic 68 (4):1145-1162.
    We generalize tools and results from first order stable theories to groups inside a simple stable strongly homogeneous model.
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  13.  12
    Invariant Version of Cardinality Quantifiers in Superstable Theories.Alexander Berenstein & Ziv Shami - 2006 - Notre Dame Journal of Formal Logic 47 (3):343-351.
    We generalize Shelah's analysis of cardinality quantifiers for a superstable theory from Chapter V of Classification Theory and the Number of Nonisomorphic Models. We start with a set of bounds for the cardinality of each formula in some general invariant family of formulas in a superstable theory (in Classification Theory, a uniform family of formulas is considered) and find a set of derived bounds for all formulas. The set of derived bounds is sharp: up to a technical restriction, every model (...)
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  14.  39
    Atlanta Marriott Marquis, Atlanta, Georgia January 7–8, 2005.Matthias Aschenbrenner, Alexander Berenstein, Andres Caicedo, Joseph Mileti, Bjorn Poonen, W. Hugh Woodin & Akihiro Kanamori - 2005 - Bulletin of Symbolic Logic 11 (3).
  15.  1
    Definable connectedness of randomizations of groups.Alexander Berenstein & Jorge Daniel Muñoz - 2021 - Archive for Mathematical Logic 60 (7):1019-1041.
    We study randomizations of definable groups. Whenever the underlying theory is stable or NIP and the group is definably amenable, we show its randomization is definably connected.
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  16. Definable groups in dense pairs of geometric structures.Alexander Berenstein & Evgueni Vassiliev - forthcoming - Archive for Mathematical Logic:1-28.
    We study definable groups in dense/codense expansions of geometric theories with a new predicate P such as lovely pairs and expansions of fields by groups with the Mann property. We show that in such expansions, large definable subgroups of groups definable in the original language \ are also \-definable, and definably amenable \-definable groups remain amenable in the expansion. We also show that if the underlying geometric theory is NIP, and G is a group definable in a model of T, (...)
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  17. Dividing in the Algebra of Compact Operators.Alexander Berenstein - 2004 - Journal of Symbolic Logic 69 (3):817-829.
    We interpret the algebra of finite rank operators as imaginaries inside a Hilbert space. We prove that the Hilbert space enlarged with these imaginaries has built-in canonical bases.
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  18.  12
    Fields with a dense-codense linearly independent multiplicative subgroup.Alexander Berenstein & Evgueni Vassiliev - 2020 - Archive for Mathematical Logic 59 (1-2):197-228.
    We study expansions of an algebraically closed field K or a real closed field R with a linearly independent subgroup G of the multiplicative group of the field or the unit circle group \\), satisfying a density/codensity condition. Since the set G is neither algebraically closed nor algebraically independent, the expansion can be viewed as “intermediate” between the two other types of dense/codense expansions of geometric theories: lovely pairs and H-structures. We show that in both the algebraically closed field and (...)
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  19.  5
    Isometry Groups of Borel Randomizations.Alexander Berenstein & Rafael Zamora - 2020 - Notre Dame Journal of Formal Logic 61 (2):297-316.
    We study global dynamical properties of the isometry group of the Borel randomization of a separable complete structure. We show that if properties such as the Rokhlin property, topometric generics, and extreme amenability hold for the isometry group of the structure, then they also hold in the isometry group of the randomization.
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  20.  11
    Supersimple Structures with a Dense Independent Subset.Alexander Berenstein, Juan Felipe Carmona & Evgueni Vassiliev - 2017 - Mathematical Logic Quarterly 63 (6):552-573.
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  21.  7
    Almost Indiscernible Sequences and Convergence of Canonical Bases.Itaï Ben Yaacov, Alexander Berenstein & C. Ward Henson - 2014 - Journal of Symbolic Logic 79 (2):460-484.
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